On the probabilistic Cauchy theory of the cubic nonlinear Schrödinger equation on $\mathbb {R}^d$, $d \geq 3$

Abstract:

We consider the Cauchy problem of the cubic nonlinear Schrödinger equation (NLS) $: i \partial _t u + \Delta u = \pm |u|^{2}u$ on $\mathbb {R}^d$, $d \geq 3$, with random initial data and prove almost sure well-posedness results below the scaling-critical regularity $s_\mathrm {crit} = \frac {d-2}{2}$. More precisely, given a function on $\mathbb {R}^d$, we introduce a randomization adapted to the Wiener decomposition, and, intrinsically, to the so-called modulation spaces. Our goal in this paper is three-fold. (i) We prove almost sure local well-posedness of the cubic NLS below the scaling-critical regularity along with small data global existence and scattering. (ii) We implement a probabilistic perturbation argument and prove ‘conditional’ almost sure global well-posedness for $d = 4$ in the defocusing case, assuming an a priori energy bound on the critical Sobolev norm of the nonlinear part of a solution; when $d \ne 4$, we show that conditional almost sure global well-posedness in the defocusing case also holds under an additional assumption of global well-posedness of solutions to the defocusing cubic NLS with deterministic initial data in the critical Sobolev regularity. (iii) Lastly, we prove global well-posedness and scattering with a large probability for initial data randomized on dilated cubes.